Landlord–Tenant Dilemma: How Does the Conflict Affect the Design of Building Energy Systems?

Abstract

To achieve climate goals, the European Union needs to increase building renovation rates. In owner-occupied buildings, energy cost savings provide financial incentives for renovation. However, 30% of all Europeans live in rented property, where conflicting stakeholder interests arise. Landlords are responsible for renovation decisions (building envelope and energy system) and the corresponding investments. Tenants face rising rents and only slightly benefit from falling energy costs. The literature calls this conflict the landlord–tenant dilemma. However, publications lack a quantification, leaving gaps in understanding its impact on technology choices and the heat transition. To address this, we incorporate the perspectives of landlords and tenants in a model-based approach for optimized technology choice (mixed-integer linear program). We compare optimal individual technology choices with the total cost optimum (including costs for landlords and tenants) for renovation decisions. Additionally, we examine how changes in the regulatory framework affect the economically driven landlord’s technology choice. Our study reveals that total costs and emissions are up to 60% and 283% higher for landlords deciding for rented houses compared to owner-occupied properties. Current approaches to solve the dilemma partly favor the development of climate-friendly energy systems. However, the renovation of the building envelope and operation costs are mostly disregarded in the decisions of landlords.

1. Introduction

The building sector is responsible for 36% of the greenhouse gas emissions in the European Union (EU) [1]. To achieve the climate goals, e.g., a climate-neutral building stock by 2050, an increase in the renovation rate of buildings is required. While the European Commission is aiming for an increase in the renovation rate from an average of 1% to 2% [2], recent studies assume a minimum renovation rate of 3% required to achieve the climate targets [3]. Apart from renovating the building envelope, a transformation of the building energy system (BES) is crucial to defossilize the heating sector since it is mostly based on fossil-fueled systems.

This transformation incurs considerable costs for renovation measures, which raises the question of who will bear the costs of the heat transition. Especially in rented property, where 30% of all Europeans live [4], conflicting stakeholder interests and the challenge of an appropriate cost distribution between the landlord and the tenant arises. Landlords are responsible for the renovation decisions on the building envelope and the energy system, as well as the corresponding investment, but will not benefit from future energy cost savings. On the other side, tenants face rising rents and often only slightly decreased energy costs. This causes the so-called landlord–tenant dilemma, which involves two major challenges concerning the landlord’s decisions:

• Lack of incentives to invest in renovation measures;
• Limited interest in renovation decisions that reduce operating costs for tenants.

Based on these issues, a third problem arises:

3.

Insufficient emission reductions and incentive to reach the climate targets.

To achieve the climate targets, it is therefore essential to resolve these challenges. This means incentivizing investment decisions by landlords that simultaneously contribute to a favorable solution for tenants and do contribute to the climate goals. The aim of this paper is therefore to demonstrate the effects of the landlord–tenant dilemma in a model-based approach and to assess the influence of the current regulatory framework.

1.1. Regulatory Framework

In comparison with other EU countries, Germany has the largest rental share in the residential building stock of 51% [4]. Therefore the current regulatory framework of Germany serves as a basis for the present study. According to the German Civil Code (BGB), landlords in Germany generally have the right to increase the rent up to a local reference rent, which is based on the average rent of comparable housing within a municipality over the past six years. Addressing the landlord–tenant dilemma, Germany has additionally introduced a retrofitting fee. This fee aims to refinance investments in renovation measures by allocating a proportion of the investment to the tenants. However, the calculation of the retrofitting fee is only dependent on the cost of the renovation and independent of the type of renovation measures [5] (see Equation (4) in Section 2.4.2). This cost-based calculation often leads to increased base rents (rent without energy costs), while tenants do not necessarily benefit from reduced energy costs [6]. Moreover, the viability of the retrofitting fee strongly depends on the development of the local reference rent, to which the landlord is always allowed to increase the rent regardless of any renovation [5]. (See Section 2.4.2 for further information on the retrofitting fee and the local reference rent).

To generally incentivize investment decisions and especially investment decisions in renovation measures favorable for the heat transition, Germany has introduced different regulations in recent years (see

Table 1. Overview of presented solutions for the landlord–tenant dilemma.

Current Regulations	Discussed in Literature
Tenant Electricity	[7]
CO2 cost allocation	[8,9]
Suggested Approach
Adjustment of the retrofitting fee	[8,10,11,12,13,14]
Energy-differentiated local reference rent	[6,8,10]
One-third model	[8,13,14,15]
Energy and climate fund model	[5,8]
Separate surcharge on base rent	[6,13]
Differentiation of subsidies by landlord type	[14]
Obligation to renovate	[14]

). In 2017, the Tenant Electricity Act came into force as part of the Renewable Energy Source Act (EEG) and the German Energy Act (EnWG). By simplifying the sale of self-generated electricity from landlords to tenants, it aims to create incentives for landlords to invest in systems for the self-generation of electricity. Furthermore, since 2023, a legal framework specifies the allocation of CO₂ costs associated with fossil-based technologies between landlords and tenants in Germany. The landlord’s share of costs increases with higher specific CO₂ emissions. The regulation aims to create incentives for landlords to invest in technologies that are in line with climate targets (see also Section 2.4.5 on tenant electricity and Section 2.4.4 on CO₂ cost allocation).

Moreover, the literature proposes a variety of further potential approaches to address the landlord–tenant dilemma (see Table 1). The most frequently cited solutions in the literature are the following:

• Adjustment of the retrofitting fee: For an adjusted retrofitting fee, the literature suggests a retrofitting fee that is no longer solely dependent on the cost, but also on energy savings. The aim is to prevent landlords from solely benefiting from increasing the costs of renovations.
• Energy-differentiated local reference rent: The energy-differentiated local reference rent pursues a similar goal by taking into account energy-related attributes of buildings within an energy-differentiated local reference rent. A higher local reference rent for climate-friendly energy systems is intended to create incentives for renovation decisions that reduce energy demand and greenhouse gas emissions and therefore, simultaneously costs.
• One-third model: The one-third model, first presented by Mellwig et al. [15], states that the costs of an energy-efficient renovation should be equally allocated between landlords, tenants, and the state.

Among the suggested solutions, the energy-differentiated local reference rent is the only proposed solution that already has a legal basis in the context of the rental law in Germany (§ 558 BGB). Moreover, current political efforts with a law to reform the rent index law indicate that local reference rents will play a crucial role in the future [16]. Hence, out of the suggested approaches from the literature, this study focuses on the energy-differentiated local reference rent.

1.2. Landlord–Tenant Dilemma in the Literature

The landlord–tenant dilemma and its potential solutions have been studied in various research disciplines, encompassing economic, legal, and social science as well as engineering approaches.

Table 2. Landlord–tenant dilemma in the literature.

Considered in publication:

Building performance evaluation

Rental law (BGB)

Other legal frameworks

Building envelope

Building energy system

Building modeling

Rent payments

Retrofitting fee

Local referencerent

Energy differentiation

Building Energy Act (GEG)

Subsidies

CO2 cost allocation

Feed-in tariffs

Tenant electricity

Economic science

2022

Ahlrichs et al. [11]

2021

Henger et al. [8]

2020

Henger et al. [5]

2019

Mellwig et al. [15]

2016

Kossmann et al. [10]

Legal science

2019

Gaßner et al. [13]

2011

Neitzel et al. [6]

2009

Ekardt et al. [12]

Social science

2022

Taruttis et al. [9]

2022

März et al. [17]

2021

Lang et al. [18]

2019

März et al. [14]

Engineering

2022

Petkov et al. [19]

2022

Braeuer et al. [7]

2015

Steinbach [20]

provides an overview of the different research disciplines and their consideration of relevant aspects related to the landlord–tenant dilemma. The evaluation reveals three relevant aspects—building performance evaluation, rental law, and other legal frameworks (GEG, subsidies, CO₂ costs, feed-in, and tenant electricity).

With regard to the building performance evaluation, the literature overview denotes whether the studies consider renovations of the building envelope and the BES along with the studies’ level of detail in the building modeling. In the area of rental law, the focus of the review is on whether rent payments and the retrofitting fee or the local reference rent and, in particular, energy-differentiated characteristics are taken into account. Furthermore, the overview lists whether requirements from the Building Energy Act (GEG), subsidies, CO₂ cost allocation, feed-in tariffs, and tenant electricity are taken into account (other legal frameworks).

Studies from economics [8,10,11,15] and legal science [6,12,13] focus on similar relevant aspects related to the landlord–tenant dilemma. They mainly neglect a precise building performance evaluation but extensively cover the current rental law. Some even take into account all listed aspects of the rental law [5,6,8,10,13]. Aspects that are summarized under other legal frameworks are mostly not included in the studies. Only Henger et al. [5,8] consider subsidies and partly CO₂ cost allocation, and Mellvig et al. [15] also account for subsidies.

In the social science analysis [9,14,17,18], the building performance evaluation is mainly disregarded, except in the work of Taruttis et al. [9], which considers renovation options for the building envelope and the BES including a simplified modeling approach. However, the studies from social science mostly show a detailed consideration of the stakeholders’ willingness to pay and some aspects of rental law (e.g., rent payments and an energy differentiation), but besides März et al. [14] they greatly simplify other aspects of the rental law (e.g., retrofitting fee and local reference rent). In addition, all considered publications from social science only occasionally address aspects of other legal frameworks (e.g., subsidies [14] or CO₂ cost allocation) [9], but neglect the holistic view of all regularities (e.g., GEG, feed-in tariffs, tenant electricity).

Engineering approaches represent actual renovation options for the building envelope and BESs with varying degrees of accuracy in building modeling [7,19,20]. However, these models widely disregard the applicable rental law and thus the individual stakeholder involved. Only Petkov et al. [19] take detailed rent payments into account, but they neglect mechanisms for cost allocation which are supposed to incentivize investment decisions for landlords (e.g., retrofitting fee, local reference rent, and energy differentiation). While the latter are partially included in Steinbach’s study [20], he neglects all aspects of other related regulatory frameworks. Petkov et al. [19] and Braeuer et al. [7] also disregard specifications from GEG, subsidies, and CO₂ cost allocation, but consider feed-in tariffs. Furthermore, only the work of Braeuer et al. [7] analyses the effect of tenant electricity on the landlord’s decision to invest in the BES.

This literature review demonstrates that scientific publications so far mostly focus on isolated topics instead of combining all relevant aspects related to the landlord–tenant dilemma. For instance, Braeuer et al. [7] examine tenant electricity in terms of its profitability for the landlord but neglect rental payments. Henger et al. [5,8] and Mellwig et al. [15] extensively address the rental law and further combine it with subsidies, but disregard a specific examination of the building and some regulations (e.g., GEG and tenant electricity). Since these topics are interdependent, the literature so far lacks a holistic investigation that considers all relevant aspects of the landlord–tenant-dilemma. Further, most publications focus on qualitative studies on the landlord–tenant dilemma and its solutions. A precise quantification of the associated technical decisions by landlords and the consequences for tenants (e.g., rent or energy costs) is not known. This further implies that the literature has not yet investigated the specific impact of the dilemma on greenhouse gas emissions and thus on the heat transition.

1.3. Contributions

To address the research gap, we develop an optimization model in form of a mixed-integer linear program (MILP). This model-based approach allows the integration of all relevant aspects concerning the landlord–tenant-dilemma and further enables a quantification of the dilemma and its potential solutions. The developed optimization model is based on an existing MILP which includes renovation measures of the building envelope and the BES [21]. We extend the model by the respective legal framework of Germany and incorporate the different perspectives of the stakeholders. With this framework, we compare individual technology choices for rented buildings with the total cost optimum and emissions of owner-occupied buildings. Furthermore, we close the current research gap by answering the following questions:

• How does the building owner’s renovation decision differ between owner-occupied and rented buildings?
• Do tenant electricity, CO₂ cost allocation, or an energy-differentiated local reference rent resolve the landlord–tenant dilemma?
• What is the impact of the landlord–tenant dilemma on the heat transition for representative use cases?

Figure 1 illustrates the structure of this study. Section 2 presents the MILP-based optimization framework. With two objective functions, the framework includes the two different perspectives of a landlord maximizing annual profits and an owner minimizing annual total costs. As a result, the optimization chooses an optimized building envelope and energy system, its operation, and, in the case of a land–tenant relationship, the rent development. In Section 3, we present a use case with two typical multi-family houses (MFHs) in Germany. Finally, Section 4.1 addresses the first research question, Section 4.2 deals with the second research question, and the third research question is discussed in both result sections (Section 4.1 and Section 4.2). The results are evaluated according to the three defined challenges of the landlord–tenant dilemma (see Section 1).

2. Method

In this section, we present the optimization framework and the mathematical formulation of the implemented regulatory framework. We extend an existing framework for design and operational optimization of residential buildings presented by Schütz et al. [21]. To achieve the best possible solution, we keep the original structure of an MILP, which is a deterministic approach and can therefore achieve optimality in contrast to heuristic methods [22]. Furthermore, studies show that linearized MILPs provide a good trade-off between the quality of the results and computational effort [23,24]. As we extend the existing MILP by Schütz et al. [21], in the following, we focus on the mathematical formulations of the modified and added equations. For a complete documentation of the initial model, we refer to [21].

2.1. Optimization Framework

The model aims to find a cost-optimal BES design and thereby decides about different modernization measures. To this end, we define various measures as decisions of the optimization model. The model encompasses both measures for the building envelope and the energy system (see Section 2.3). Through mathematic optimization, we determine the BES with the lowest life cycle cost.

Figure 2 presents an overview of the optimization framework. While the original model [21] aims to find the cost-optimal solution for the design and operation of the BES, we add the perspective of a landlord to the model. We assume that the modernization decision of the landlord is economically driven. Thus, we implement a new objective function that assesses the income of the landlord under different regulatory frameworks. The income of the landlord depends in particular on rental income, while the permissible level of rental payments is significantly influenced by the regulatory framework. The comparison of the design optimization for both objective functions, cost-optimal in terms of overall costs and cost-optimal for the landlord, allows conclusions regarding the steering impact of the examined regulatory framework on the modernization decision (see Section 2.2 for detailed information on the objective functions).

Besides the decision on the BES, including the building envelope and the energy system, the optimization framework also decides on the optimal energy system operation. In case of the maximized annualized profit of a landlord, the model also includes decisions on the rent development in the allowed legal range (see Section 2.3 for further information).

In the first box, Figure 2 shows the basic components of the optimization framework, including the modeling of the building and the energy system, and the regulatory framework. Building data, weather data, and precalculated domestic hot water and electrical profiles, as well as techno-economic and ecologic data serve as input for the optimization. For the parameterization of the use case, we refer to Section 3. To calculate the thermal energy demand for room heating, we use a 5R1C demand model according to DIN EN ISO 13790 [25] (see Section 2.4.1). Moreover, the current German regulatory framework determines the formulation of the rental constraints presented in Section 2.4.2.

2.2. Objective Functions

The objective of the model is to find a cost-optimal technology choice at the beginning of a specified time period from different perspectives. The annuity serves as an economic metric [26]. We distinguish two different objective functions: the maximum annualized profit of a landlord of a rented building (see Equation (1)) and the minimum annualized costs of a building owner (see Equation (2)).

Table 3. Overview of the costs of the objective function.

Category		Stakeholder
	Owner-Occupied	Landlord	Tenant
Investment	−	−
Installation	−	−
Maintenance	−	−
Consumption	−	−	−
Emissions	−	−	−
Metering	−		−
Feed-in/self-consumption surcharges	+	+
Subsidies	+	+
Rent		+	−
Tenant electricity *		(+)	(−)

* only if tenant electricity is applied.

illustrates the relevant financial contributions for the respective stakeholders. Costs are marked with a minus sign (−), revenues with a plus sign (+), and no consequences with an empty field ( ). Investments, installation costs, and subsidies are incurred at the beginning of the period under consideration. Maintenance costs, consumption costs, emission costs, metering costs, feed-in revenues, rental payments, and tenant electricity payments are incurred annually. A price-dynamic present value factor is determined to reflect price changes, except for rental payments.

The annualized profit of a landlord (see Equation (1)) consists of different revenues minus expenditures. The landlord acquires incomes from rental payments $c^{\mathrm{rent}}$, tenant electricity $c^{\mathrm{TEL}}$, electricity fed into the grid $c^{\mathrm{feed}}$, and subsidies for modernization measures $c^{\mathrm{sub}}$. The expenditures consist of capital expenditures (CAPEX) $c^{\mathrm{inv}}$ and operational expenditures (OPEX). The OPEX further divides into operation and maintenance costs $c^{\mathrm{o}\&\mathrm{m}}$, demand-related costs $c^{\mathrm{dem}}$, metering costs $c^{\mathrm{met}}$, and emission costs $c_{\mathrm{landlord}}^{{\mathrm{CO}}_2}$.

$$max{c}^{\mathrm{total}}=c^{\mathrm{rent}}+c^{\mathrm{TEL}}-c^{\mathrm{inv}}-c^{\mathrm{inst}}-c^{\mathrm{o}\&\mathrm{m}}-c^{\mathrm{dem}}-c^{\mathrm{met}}-c_{\mathrm{landlord}}^{{\mathrm{CO}}_2}+c^{\mathrm{feed}}+c^{\mathrm{sub}}$$

(1)

The minimum annualized costs of the owner consist of CAPEX and OPEX (see Equation (2)). The objective function originates from the work of Schütz et al. [21] and, in contrast to the landlord’s profit, rental payments and incomes of tenant electricity are missing.

$$min{c}^{\mathrm{total}}=c^{\mathrm{inv}}+c^{\mathrm{inst}}+c^{\mathrm{o}\&\mathrm{m}}+c^{\mathrm{dem}}+c^{\mathrm{met}}+c^{{\mathrm{CO}}_2}-c^{\mathrm{feed}}-c^{\mathrm{sub}}$$

(2)

2.3. Decision Variables

As a result of the optimization, the model decides on the cost-optimal combination of the building envelope and the energy system design and operation. Thereby, we consider various modernization measures $dev\in DEV$, which can be combined in any way. Regarding the building envelope, we distinguish the modernization of the facade, the roof, and the windows. For each component, four levels with increasing insulation standards can be selected independently. In the following, we designate the different standards with the indices 0 for no modernization measure, 1 for moderate modernization, 2 for more advanced modernization, and 3 for an ambitious modernization. For each standard, we model different costs and U values (see Appendix A.1). Regarding the energy system, we consider boilers (BOI), combined heat and power engines (CHP), air source heat pumps (HP), electric heaters (EH), solar thermal collectors (STC), photovoltaics (PV), thermal energy storage (TES), and batteries (BAT). The capacities of each heater and storage system are modeled as a discrete choice, while the area of solar collectors is modeled as a continuous variable. In case of optimized costs in a landlord–tenant relation, the MILP also determines how the rent can be increased according to regulations. For the rent development, two rent increase mechanisms, increase to the local reference rent and the application of the retrofitting fee, are available (see Section 2.4).

2.4. Optimization Framework

The following section describes the basic structure of the optimization framework, including the modeling of the building and the energy system as well as the regulatory boundaries.

2.4.1. Building and Energy System

The building and energy system models are provided by the existing MILP. The thermal behavior of the building is described by a 5R1C model based on DIN EN ISO 13790 [25]. The 5R1C model summarizes the entire building into one thermal capacity and five thermal resistances. The whole building is modeled as one thermal zone with central heating. Building data for the initial building standard are taken from TABULA typologies [27]. To determine the size of the initial energy system, the standard heating load according to DIN EN 12831-1 is used with a set indoor temperature of 20 °C [28]. The modeling of the 5R1C model is described in [21]. We describe the parameterization of the use case in Section 3.

2.4.2. Rental Payments

In the following section, we describe the modeling of rental payments under current law. The cost for tenants for each year (y) are obtained by multiplying variable specific rental payments $c_y^{\mathrm{rent}}$ with the living area (A) and discounting it to the time of consideration by the discounting factor q. The annualized rental payments $c_{\mathrm{ann}}^{\mathrm{rent}}$ are then determined by applying the capital recovery factor (${\mathrm{f}}_{\mathrm{CRF}}$) to the sum of yearly rent payments $c_y^{\mathrm{rent}}$ over the considered time period T (see Equation (3)).

$$c^{\mathrm{rent}}=\left(\sum _{y\in (1,\dots ,T)}\frac{c_y^{\mathrm{rent}}·12·\mathrm{A}}{q^y}\right)·{\mathrm{f}}_{\mathrm{CRF}}$$

(3)

In Germany, the rental law provides regulations to limit rent increase, specifically the allowed rent increase mechanisms for the local reference rent according to § 558 BGB and for the retrofitting fee according to § 559 BGB. The combination of both mechanisms is integrated based on a German Federal Court of Justice ruling [29]. For both mechanisms, we only consider existing tenants, as more extensive regulations apply to new tenants (e.g., rent control—see § 556d BGB). The retrofitting fee enables the landlord to pass 8% of the costs of renovation measures to the tenants, comprising of investments $c_{dev}^{\mathrm{inv},\mathrm{ann}}$ and installation costs $c_{dev}^{\mathrm{inst},\mathrm{ann}}$. Once applied, the retrofitting fee becomes part of the newly determined rent. The retrofitting fee has absolute caps depending on the rent before the renovation. Equation (4) describes the calculation of the monthly retrofitting fee $c_y^{\mathrm{RF}}$, which refers to the living area A of the building.

$$c_y^{\mathrm{RF}}·12·A\le 0.08·(\sum _{dev\in DEV}(c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})-c_{dev}^{\mathrm{sub}})/{\mathrm{f}}_{\mathrm{CRF}}$$

(4)

The local reference rent, which is determined every two years, is specified in rent indices and reflects the rent for comparable residential spaces. The landlord may increase the rent in accordance with the housing characteristics, in particular the location, size, type, features, and quality. The rent increase to the local reference rent has relative capping limits depending on the housing market situation. Once applied, the reference rent becomes part of the newly determined rent. A further increase to the local reference rent can only take place again if the rent falls below the local reference rent which is adjusted every two years on the basis of the last six years.

The review of the current literature reveals an energy-differentiated local reference rent as one promising solution for the landlord–tenant dilemma (see Section 1.1). To evaluate the impact, we implement a local reference rent with and without energy differentiation. In Germany, there are two different approaches for energy-differentiated local reference rents, either characterizing the energetic quality of a building by specific energy key performance indicators, such as the final energy demand of a building, or individual characteristics relevant for the energy consumption of a building [30,31]. For the latter, local authorities use different characteristics, such as the year of installation of the heating system, the type of windows, or the energy quality of the building envelope. In Germany, Hamburg in particular is regarded as a positive example of the implementation of an energy-differentiated local reference rent [32]. Hamburg’s rent index defines the energy-differentiated local reference rent based on the final energy demand for heating and domestic hot water as defined for the building energy certificate (§§ 79–88 GEG). The final energy demand must be determined after renovation measures and, in contrast to individual characteristics, it allows for a standardized comparison of buildings. Therefore, in this work, we implement the energy-differentiated local reference rent following the rent index of Hamburg. The binary variable $x_{lev}$ indicates which local reference rent level ($lev$) has to be applied based on the achieved final energy demand $Q_{\mathrm{tot}}^{\mathrm{fin}}$ (see Equation (5)). Thereby, each energy quality level has a maximum final energy demand $Q_{lev}^{\mathrm{fin}}$ and only one energy level can be chosen (see Equation (7)). The corresponding local reference rent $c_y^{\mathrm{lrr}}$ is selected from the available local reference rents ${\mathrm{c}}_{y,lev}^{\mathrm{lrr}}$ (see Equation (6)). The variable $x_y^{\mathrm{switch}}$ indicates whether the local reference rent is applied in the respective year if the retrofitting fee was previously used. The switch is only possible once since the retrofitting fee becomes part of the newly determined rent (8).

$$Q_{\mathrm{tot}}^{\mathrm{fin}}\le \sum _{lev\in L}{\mathrm{Q}}_{lev}^{\mathrm{fin}}·x_{lev}$$

(5)

$$c_y^{\mathrm{lrr}}=\sum _{lev\in L}{\mathrm{c}}_{y,lev}^{\mathrm{lrr}}·x_{lev}\forall y\in \{1,\dots ,T\}$$

(6)

$$\sum _{lev\in L}{x}_{lev}\le 1$$

(7)

The landlord must decide whether to apply the reference rent to the existing rent before modernization or whether to increase the rent to the comparative rent of the modernized condition. To model this decision, we introduce the binary $x_y^{\mathrm{switch}}$. This variable indicates whether the local reference rent is applied in the respective year if the retrofitting fee was previously used. The switch is only possible once since the retrofitting fee becomes part of the newly determined rent (8). If the reference rent is applied by setting $x_y^{\mathrm{switch}}$ to zero, the base rent is the rent of the previous year $c_0^{\mathrm{rent}}$ (year 0 before the start of the period under consideration) or the initial local reference rent of the unrenovated state $c_{1,lev}^{\mathrm{lrr},\mathrm{init}}$ and the reference rent is charged on top (see Equation (9)). Alternatively, the rent can be increased according to the local reference rent of the refurbished condition by setting $x_y^{\mathrm{switch}}$ to one in Equation (10).

$$x_y^{\mathrm{switch}}\le x_{y+1}^{\mathrm{switch}}\forall y\in \{1,\dots ,T-1\}$$

(8)

$$c_y^{\mathrm{rent}}\le c^{\mathrm{rf}}+max(c_0^{\mathrm{rent}},c_{1,lev}^{\mathrm{lrr},\mathrm{init}})+M·x_y^{\mathrm{switch}}\forall y\in \{1,\dots ,T\}$$

(9)

$$c_y^{\mathrm{rent}}\le c_y^{\mathrm{lrr}}+M·(1-x_y^{\mathrm{switch}})\forall y\in \{1,\dots ,T\}$$

(10)

For both rent increase mechanisms, independent caps must be respected. The cap for the local reference rent consists of a maximum percentage rent increase within three years (§ 558 Art. 3 BGB). The capping percentage $c_{\mathrm{limit}}^{\mathrm{lrr}}$ depends on the local housing market (11). The cap of the reference rent consists of an absolute rent increase value per ${\mathrm{m}}^2$ (§ 559 Art. 3a BGB). The cap $c_{\mathrm{limit}}^{\mathrm{rf}}$ is dependent on the rent before the application of the reference rent (see Equation (12)) and is constrained by Equation (13). With Equations (14) and (15), we take into account the independence of both capping limits.

$${\mathrm{c}}_{\mathrm{limit}}^{\mathrm{lrr}}=\left\{\begin{array}{cc}0.15\hfill & \mathrm{if}\mathrm{tense}\mathrm{market}\hfill \\ 0.20\hfill & \mathrm{if}\mathrm{no}\mathrm{tense}\mathrm{market}\hfill \end{array}\right.$$

(11)

$${\mathrm{c}}_{\mathrm{limit}}^{\mathrm{rf}}=\left\{\begin{array}{cc}2\frac{€}{{\mathrm{m}}^2}\hfill & \mathrm{if}max(c_0^{\mathrm{rent}},c_{1,lev}^{\mathrm{lrr},\mathrm{init}})\le 7\frac{€}{{\mathrm{m}}^2}\hfill \\ 3\frac{€}{{\mathrm{m}}^2}\hfill & \mathrm{if}max(c_0^{\mathrm{rent}},c_{1,lev}^{\mathrm{lrr},\mathrm{init}})>7\frac{€}{{\mathrm{m}}^2}\hfill \end{array}\right.$$

(12)

$$c^{\mathrm{rf}}\le {\mathrm{c}}_{\mathrm{limit}}^{\mathrm{rf}}$$

(13)

$$c_y^{\mathrm{rent}}\le c^{\mathrm{rf}}+{\mathrm{c}}_{\mathrm{limit}}^{\mathrm{lrr}}·c_{y-3}^{\mathrm{rent}}+M·x_y^{\mathrm{switch}}\forall y\in \{1,\dots ,T\}$$

(14)

$$c_y^{\mathrm{rent}}\le {\mathrm{c}}_{\mathrm{limit}}^{\mathrm{lrr}}·c_{y-3}^{\mathrm{rent}}+M·(1-x_y^{\mathrm{switch}})\forall y\in \{1,\dots ,T\}$$

(15)

2.4.3. Subsidization of Modernization Measures

In this study, we consider current German subsidy schemes according to the federal funding for energy-efficient buildings (BEG) [33]. The funding scheme distinguishes subsidies for the application of single modernization measures (BEG single measures) and the achievement of standards regarding the overall building efficiency. The BEG single measures subsidize the installation of energy-efficient heating systems and the insulation of facade components with a device-dependent subsidy share $s_{dev}^{\mathrm{BEG} \mathrm{EM}}$ (see Equation (16)). In the modeling, we only consider heating and solar devices that fulfill the requirements of BEG. If a fossil-based heating system is replaced by an HP or STC, a further 10% subsidy is granted. We make use of a binary variable $x^{\mathrm{fossilRepl}}$ to model this behavior in Equations (17)–(19). According to the German Federal Ministry for Economics and Climate Action [33], the sum of subsidies $c_{dev}^{\mathrm{sub},\mathrm{BEG} \mathrm{EM}}$ over all devices must not exceed 60,000 € per apartment, which is ensured by Equation (20).

$$c_{dev}^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}\le (c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})·s_{dev}^{\mathrm{BEG} \mathrm{EM}}\forall dev\in DEV\setminus \left\{\mathrm{HP},\mathrm{STC}\right\}$$

(16)

$$\sum _{dev}{x}_{dev,i}\le (1-x^{\mathrm{fossilRepl}})·2\forall dev\in \left\{\mathrm{CHP},\mathrm{BOI}\right\}$$

(17)

$$c_{dev}^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}\le s_{dev}^{\mathrm{BEG} \mathrm{EM}}·(c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})+M·x^{\mathrm{fossilRepl}}\forall dev\in \left\{\mathrm{HP},\mathrm{STC}\right\}$$

(18)

$$c_{dev}^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}\le (s_{dev}^{\mathrm{BEG} \mathrm{EM}}+0, 1)·(c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})+M·(1-x^{\mathrm{fossilRepl}})\forall dev\in \left\{\mathrm{HP},\mathrm{STC}\right\}$$

(19)

$$c^{\mathrm{sub},\mathrm{BEG} \mathrm{EM}}\le \sum _{dev}{c}_{dev}^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}\le {\mathrm{f}}_{\mathrm{CRF}}·n^{Apart}·\mathrm{60,000}\forall dev\in DEV$$

(20)

Modernization measures on the building envelope must comply with maximum U values $U_{dev}^{\mathrm{req}, \mathrm{BEG} \mathrm{EM}}$ to grant subsidies. We make use of the binary variable $x_{dev}^{\mathrm{BEG} \mathrm{EM}}$ to check the eligibility of each modernization measure for $dev$$ϵ$ {facade, roof, window} and calculate the subsidies for shell measures as follows:

$$\begin{array}{cc}{c}_{dev}^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}\le s_{dev}^{\mathrm{BEG} \mathrm{EM}}·(c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})·x_{dev}^{\mathrm{BEG} \mathrm{EM}}\hfill & \hfill \forall dev\in \left\{\mathrm{facade},\mathrm{roof},\mathrm{window}\right\}\end{array}$$

(21)

$$U_{\mathrm{dev}}\le U_{\mathrm{dev}}^{\mathrm{req}, \mathrm{BEG} \mathrm{EM}}+M·(1-x_{dev}^{\mathrm{subElig}, \mathrm{BEG} \mathrm{EM}})\forall dev\in \left\{\mathrm{facade},\mathrm{roof},\mathrm{window}\right\}$$

(22)

$$U_{\mathrm{dev}}\geqq U_{\mathrm{dev}}^{\mathrm{req}, \mathrm{BEG} \mathrm{EM}}-M·x_{dev}^{\mathrm{subElig}, \mathrm{BEG} \mathrm{EM}}\forall dev\in \left\{\mathrm{facade},\mathrm{roof},\mathrm{window}\right\}$$

(23)

If the requirements for primary energy demand and transmission heat loss are met after modernization, subsidies are granted in accordance with BEG WG. The subsidy rate is graduated according to the achieved efficiency class $class\in C^{BEGWG}$ depending on the yearly primary energy demand and the transmission heat loss coefficient. The set of efficiency classes C contains the efficiency house standards (EHS) {EHS85, EHS70, EHS40, EHS85RE, EHS70RE, EHS40RE, EHS85REWPB, EHS70REWPB, EHS40REWPB} named after the specific primary energy demand relative to the heated floor area. If an efficiency class is reached and the heating demand is covered by a share of at least 65% renewable energy, the efficiency class is part of the set of renewable energy (RE) efficiency classes $C_{\mathrm{RE}}^{\mathrm{BEG}\mathrm{WG}}$ and higher subsidies are granted. The set $C_{\mathrm{RE}}^{BEGWG}$ contains the classes {EHS85RE, EHS70RE, EHS40RE}. Furthermore, the BEG defines worst-performing buildings (WPB) with a final energy demand higher than 250 $\mathrm{kWh}/\left({\mathrm{m}}^2\mathrm{a}\right)$. If the modernization of a building belonging to this category prior retrofit meets the requirements of the efficiency classes, the efficiency class is part of the set $C_{\mathrm{WPB}}^{\mathrm{BEG}\mathrm{WG}}$ and a further bonus is granted. Thereby, the classes {EHS85REWPB, EHS70WPB, EHS40WPB} form the set $C_{WPB}^{BEGWG}$. For each efficiency class $c\in C^{\mathrm{BEG}\mathrm{WG}}$, we define a binary variable $x_{class}^{BEGWG}$ indicating whether the requirements of each class are fulfilled. For each class, the subsidy share $s_{class}^{\mathrm{BEG}\mathrm{WG}}$ is granted. Considering the maximum subsidy amounts, we model the BEG WG subsidies with the following constraints in Equations (24)–(26). Only one $class\in C^{\mathrm{BEG}\mathrm{WG}}$ may be selected and only if this was only achieved as a result of the modernization (see Equation (27)). Equation (28) ensures that the binary variable $x^{\mathrm{retro}}$ equals one only if at least one $dev\in DEV$ was retrofitted.

$$\frac{c_{class}^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}}{s_{class}^{\mathrm{BEG} \mathrm{WG}}}\le {\mathrm{f}}_{\mathrm{CRF}}·n^{\mathrm{Apart}}·\mathrm{120,000}€\forall class\in C^{\mathrm{BEG} \mathrm{WG}}\setminus C_{\mathrm{RE}}^{\mathrm{BEG} \mathrm{WG}}$$

(24)

$$\frac{c_{class}^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}}{s_{class}^{\mathrm{BEG} \mathrm{WG}}}\le {\mathrm{f}}_{\mathrm{CRF}}·n^{\mathrm{Apart}}·\mathrm{150,000}€\forall class\in C_{\mathrm{RE}}^{\mathrm{BEG} \mathrm{WG}}$$

(25)

$$c_{class}^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}\le \sum _{dev}{\mathrm{f}}_{\mathrm{CRF}}·s_{class}^{\mathrm{BEG} \mathrm{WG}}·(c_{dev}^{\mathrm{inv}}+c_{dev}^{\mathrm{inst}})\forall class\in C^{\mathrm{BEG} \mathrm{WG}}$$

(26)

$$\sum _{class\in C^{\mathrm{BEG} \mathrm{WG}}}{x}_{class}^{\mathrm{BEG} \mathrm{WG}}\le x^{\mathrm{retro}}$$

(27)

$$x^{\mathrm{retro}}\le \sum _{dev}{c}_{dev}^{\mathrm{inv}}·M$$

(28)

For each class, it is checked whether the respective limits for primary energy demand $Q_{\mathrm{P}}$ and for transmission losses $H_{\mathrm{T}}$ are complied with (Equations (29) and (30)). The parameters $s_{class}^{{\mathrm{Q}}^{\mathrm{req}}}$ and $s_{class}^{{\mathrm{H}}^{\mathrm{req}}}$ contain the permissible shares of the primary energy demand and the heat transmission losses in relation to the reference building. Equation (31) ensures that funding is only granted if the limit values are complied with.

$$Q_{\mathrm{P}}\le s_{class}^{{\mathrm{Q}}^{\mathrm{req}}}·Q_{\mathrm{P}}^{\mathrm{ref}}+M·(1-x_{class}^{\mathrm{BEG}\mathrm{WG}})\forall class\in C^{\mathrm{BEG}\mathrm{WG}}$$

(29)

$$H_{\mathrm{T}}\le s_{class}^{{\mathrm{H}}^{\mathrm{req}}}·H_{\mathrm{T}}^{\mathrm{ref}}+M·(1-x_{class}^{\mathrm{BEG}\mathrm{WG}})\forall class\in C^{BEGWG}$$

(30)

$$c_{class}^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}\le x_{class}^{\mathrm{BEG}\mathrm{WG}}·M\forall class\in C^{\mathrm{BEG}\mathrm{WG}}$$

(31)

Equation (32) determines whether a renewable energy share of at least 65% of the total heating demand (space heating ($Q_{\mathrm{SH},d,t}$) and domestic hot water ($Q_{\mathrm{DHW},d,t}$)) is achieved. Here, we consider STC, HP, and EH as renewable heat generators. Solar energy and ambient heat are taken into account as renewable energy sources. Thus, non-renewable imported grid electricity must be deducted from the heat output of HP and EH. This is expressed by the auxiliary variable $P_{\mathrm{grid},\mathrm{heat},d,t}$, which is calculated in Equation (33) by subtracting the household electricity consumption $P_{\mathrm{house},d,t}$ and the current input of the BAT $P_{\mathrm{BAT},d,t}^{\mathrm{ch}}$ from the total electricity drawn from the grid $P_{\mathrm{grid},d,t}$. Further, Equation (34) checks whether the specific final energy demand in the initial state $Q_{\mathrm{End}}^{\mathrm{init}}$ exceeds the limit value of 250 kWh/(m²a) and thus decides on the assignment to the $class$∈$C_{\mathrm{WPB}}^{\mathrm{BEG} \mathrm{WG}}$.

$$\begin{array}{c}\sum _t\sum _d(Q_{\mathrm{SH},d,t}+Q_{\mathrm{DHW},d,t})·0, 65\le \sum _d\sum _t(Q_{\mathrm{STC},d,t}+Q_{\mathrm{HP},d,t}+Q_{\mathrm{EH},d,t}-P_{\mathrm{grid},\mathrm{heat},d,t})\hfill \\ \hfill +M·(1-x_{class}^{\mathrm{BEG}\mathrm{WG}})\forall class\in C_{\mathrm{RE}}^{\mathrm{BEG}\mathrm{WG}}\end{array}$$

(32)

$$P_{\mathrm{grid}, \mathrm{heat},d,t}=max(P_{\mathrm{grid},d,t}-P_{\mathrm{house},d,t}-P_{\mathrm{BAT},d,t}^{\mathrm{ch}},0)\forall d,t$$

(33)

$$x_{class}^{\mathrm{BEG} \mathrm{WG}}·250\mathrm{k}\mathrm{W}\mathrm{h}/\left({\mathrm{m}}^2\mathrm{a}\right)\le Q_{\mathrm{End}}^{\mathrm{init}}/A_{\mathrm{f}}\forall class\in C_{\mathrm{WPB}}^{\mathrm{BEG} \mathrm{WG}}$$

(34)

The total annualized subsidy amount from BEG WG can be expressed by Equation (35). The subsidy can be granted either according to BEG WG or EM, but not both. We use the binary variable $x_{choose}^{\mathrm{BEG}}$ to decide which BEG subsidy scheme is applied and model the total subsidies according to Equations (36) and (37):

$$c^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}\le \sum _{class\in c^{\mathrm{BEG} \mathrm{WG}}}{c}_{class}^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}$$

(35)

$$c_{\mathrm{total}}^{\mathrm{sub}}\le c^{\mathrm{sub}, \mathrm{BEG} \mathrm{EM}}+M·(1-x_{\mathrm{choose}}^{\mathrm{BEG}})$$

(36)

$$c_{\mathrm{total}}^{\mathrm{sub}}\le c^{\mathrm{sub}, \mathrm{BEG} \mathrm{WG}}{sub}_{class}^{\mathrm{BEG} \mathrm{WG}}+M·x_{\mathrm{choose}}^{\mathrm{BEG}}$$

(37)

2.4.4. CO₂ Cost Allocation

Since 2023, the ${\mathrm{CO}}_2$ costs of fossil-based energy solutions are split between landlord and tenant, based on a distribution scheme considering the specific emissions [34], which is illustrated in Figure 3. The higher the ${\mathrm{CO}}_2$ emissions relative to the heated living area, the higher the landlord’s share of the costs.

To estimate the cost of emissions $c^{{\mathrm{CO}}_2}$, the emissions $E_{\mathrm{tot}}$ related to the gas consumption of the energy system are relevant. These are the emissions caused by the BOI $E_{\mathrm{tot}}^{\mathrm{BOI}}$ and the CHP $E_{\mathrm{tot}}^{\mathrm{CHP}}$ (see Equation (38)). In the case of the $c^{{\mathrm{CO}}_2}$ cost allocation, the landlord always has to pay for the emissions of the gas consumption related to the electricity generation of the CHP. Therefore, the amount of emissions relevant for the allocation $E_{\mathrm{tot}}^{\mathrm{alloc}}$ is given in Equation (39), where the thermal efficiency of the CHP ${\eta }_{\mathrm{th}}$ is taken into account.

$$E_{\mathrm{tot}}=E_{\mathrm{tot}}^{\mathrm{BOI}}+E_{\mathrm{tot}}^{\mathrm{CHP}}$$

(38)

$$E_{\mathrm{tot}}^{\mathrm{alloc}}=E_{\mathrm{tot}}^{\mathrm{BOI}}+E_{\mathrm{tot}}^{\mathrm{CHP}}·{\eta }_{\mathrm{th}}$$

(39)

For owner-occupied buildings, Equation (40) describes the calculation of the total emission costs $c^{{\mathrm{CO}}_2}$, taking into account the specific CO₂ price $p^{{\mathrm{CO}}_2}$, the price-dynamic cash value factor ${\mathrm{f}}_{\mathrm{CVF}}^{{\mathrm{CO}}_2}$ of the CO₂ price, describing the price development of CO₂, and the capital recovery factor (${\mathrm{f}}_{\mathrm{CRF}}$).

$$c^{{\mathrm{CO}}_2}=E_{\mathrm{tot}}·p^{{\mathrm{CO}}_2}·{\mathrm{f}}_{\mathrm{CVF}}^{{\mathrm{CO}}_2}·{\mathrm{f}}_{\mathrm{CRF}}$$

(40)

The CO₂ cost allocation consists of 10 discrete $steps\in S^{{\mathrm{CO}}_2}$ with corresponding emission levels ${\mathrm{E}}_{step}$. In order to determine the applicable step, we define a binary variable $x_{step}^{{\mathrm{CO}}_2}$ and a continuous auxiliary variable $E_{step}^{\mathrm{aux}}$ for each $step$ out of ${\mathrm{S}}^{{\mathrm{CO}}_2}$. With Equation (41), the emissions to be allocated are assigned to each $step$ with the corresponding emission level $E_{step}$. Equation (42) states that only one emission level may be active. Further, the allocated emissions must correspond to the sum of the auxiliary variable $E_{step}^{\mathrm{aux}}$ (see Equation (43)).

$$E_{step}^{\mathrm{aux}}\le {\mathrm{E}}_{step}·x_{step}^{{\mathrm{CO}}_2}\forall step\in {\mathrm{S}}^{{\mathrm{CO}}_2}$$

(41)

$$\sum _{step\in {\mathrm{S}}^{{\mathrm{CO}}_2}}{x}_{step}^{{\mathrm{CO}}_2}\le 1$$

(42)

$$E_{\mathrm{tot}}^{\mathrm{alloc}}=\sum _{step\in {\mathrm{S}}^{{\mathrm{CO}}_2}}{E}_{step}^{\mathrm{aux}}$$

(43)

This results in the annualized emission costs to be paid by the landlord $c_{\mathrm{landlord}}^{{\mathrm{CO}}_2}$ using Equation (44), in which ${\mathrm{y}}_{step}^{{\mathrm{CO}}_2}$ represents the landlord’s share of costs per step. $E_{\mathrm{tot}}-E_{\mathrm{tot}}^{\mathrm{alloc}}$ is the part of the emission cost resulting from the electricity generation of the CHP that is always the landlord’s responsibility. Further, $p^{{\mathrm{CO}}_2}$ describes the CO₂ price and d the dynamic present value factor. Finally, the emission costs of the tenant $c_{\mathrm{tenant}}^{{\mathrm{CO}}_2}$ are determined with Equation (45).

$$c_{\mathrm{landlord}}^{{\mathrm{CO}}_2}=(\sum _{step\in {\mathrm{S}}^{{\mathrm{CO}}_2}}\left(E_{step}^{\mathrm{aux}}·{\mathrm{y}}_{step}^{{\mathrm{CO}}_2}\right)+(E_{\mathrm{tot}}-E_{\mathrm{tot}}^{\mathrm{alloc}}))·p^{{\mathrm{CO}}_2}·d·{\mathrm{f}}_{\mathrm{CRF}}$$

(44)

$$c_{\mathrm{tenant}}^{{\mathrm{CO}}_2}=c^{{\mathrm{CO}}_2}-c_{\mathrm{landlord}}^{{\mathrm{CO}}_2}$$

(45)

2.4.5. Tenant Electricity

The sale of PV electricity from the landlord to tenants is possible under the tenant electricity regulations of the EEG and the EnWG and requires an agreement between the tenant and landlord. In the case of tenant electricity contracts, the landlord acts as an electricity provider and supplies the tenants with electricity at a maximum price of 90% of the respective basic supplier tariff (§ 42a EnWG). According to EEG, tenant electricity is subsidized by a remuneration depending on the amount of generated electricity. Thus, the income $c^{\mathrm{TEL}}$ in the case of tenant electricity is the sum of the sales to the tenants $c_{\mathrm{sale}}^{\mathrm{TEL}}$ and the revenues $c_{\mathrm{rev}}^{\mathrm{TEL}}$ (Equation 46). The landlord’s income from the provision of tenant electricity is determined at a price of 90% of the basic supplier tariff $p_{\mathrm{EL}}$ (§ 42a Abs. 4 EnWG) according to Equations (47) and (48). It is assumed that the electricity provided by CHPs is sold at the same price as PV electricity.

$$c^{\mathrm{TEL}}=c_{\mathrm{sale}}^{\mathrm{TEL}}+c_{\mathrm{rev}}^{\mathrm{TEL}}$$

(46)

$$c_{\mathrm{sale}}^{\mathrm{TEL}}\le {\mathrm{f}}_{\mathrm{CRF}}·{\mathrm{f}}_{\mathrm{CVF}}^{\mathrm{el}}·\sum _t\sum _d·p^{\mathrm{el}}·0,9·(P_{\mathrm{house},d,t}+\sum _{dev\in \left\{\mathrm{HP},\mathrm{EH},\mathrm{STC}\right\}}{P}_{\mathrm{dev},d,t})$$

(47)

$$c_{\mathrm{sale}}^{\mathrm{TEL}}\le {\mathrm{f}}_{\mathrm{CRF}}·{\mathrm{f}}_{\mathrm{CVF}}^{\mathrm{el}}·\sum _t\sum _d·p^{\mathrm{el}}·0,9·P_{\mathrm{PV},d,t}^{\mathrm{use}}$$

(48)

The amount of remuneration is power-dependent and therefore differs in different time steps. According to § 21 EEG, the remuneration is only granted if the electricity has been supplied to and consumed by an end consumer. In accordance with Braeuer et al. (2022) [7], this does not include HP, EH, and BAT. Consequently, we introduce an auxiliary variable $P_{\mathrm{TEL},d,t}^{\mathrm{rev}}$ to define the PV electricity that qualifies for the surcharge in Equation (49).

$$P_{\mathrm{TEL},d,t}^{\mathrm{rev}}=max(P_{\mathrm{PV},d,t}^{\mathrm{use}}-\sum _{dev\in \left\{\mathrm{HP},\mathrm{EH}\right\}}{P}_{dev,d,t}-P_{\mathrm{BAT},d,t}^{\mathrm{ch}},0)\forall d,t$$

(49)

§ 21 EEG defines different classes for the power fed into the grid. The higher the power fed into the grid, the less remuneration is granted. Assuming a peak power smaller than 100 kWp in residential buildings, three different classes $class\in C^{\mathrm{TEL}}$ result (see Equations (50)–(52). We define the continuous variables $P_{\mathrm{PV},d,t,class}^{\mathrm{feed}}$ that determine the power in the respective class for every time step. The total PV feed-in is a combination of the classes used (see Equation (53)).

$$P_{\mathrm{TEL},d,t,10}^{\mathrm{rev}}\le 10\mathrm{k}\mathrm{W}$$

(50)

$$P_{\mathrm{TEL},d,t,40}^{\mathrm{rev}}\le 40\mathrm{k}\mathrm{W}$$

(51)

$$P_{\mathrm{TEL},d,t,100}^{\mathrm{rev}}\le 100\mathrm{k}\mathrm{W}$$

(52)

$$\sum _{class\in C_{\mathrm{TEL}}}{P}_{\mathrm{TEL},d,t,class}^{\mathrm{rev}}\le P_{\mathrm{TEL},d,t}^{\mathrm{rev}}\forall d,t$$

(53)

The income from the tenant electricity remuneration is calculated by summing the product of the PV power fed into the grid with the time step and the tenant electricity enumeration across all points in time. The parameter $b_{\mathrm{const}}$ represents the present value factor, while the parameter $pric{e}_{\mathrm{PV} \mathrm{TEL},class}$ represents the remuneration of the specific class.

$$c_{\mathrm{rev}}^{\mathrm{TEL}}\le {\mathrm{f}}_{\mathrm{CRF}}·{\mathrm{f}}_{\mathrm{CVF}}^{{\mathrm{CO}}_2}·\sum _t\sum _d·\sum _{class\in C^{\mathrm{TEL}}}{P}_{\mathrm{TEL},d,t,class}^{\mathrm{rev}}·pric{e}_{\mathrm{PV} \mathrm{TEL},class}$$

(54)

2.4.6. Feed-In Remuneration

Owners can profit from the feed-in and the self-consumption of PV and CHP power [35,36,37]. The mathematical formulations for the subsidies of PV and CHP power are presented in [21]. In this work, we only update the parameters according to current regulations. According to the EEG, PV feed-in is remunerated for 20 years (§ 25 Art. 1 EEG) depending on the installed capacity (§ 48 Art. 2 EEG). CHP feed-in is regulated by the CHP Act (KWKG) and remunerated for 30,000 full load hours (§ 8 Art. 1 KWKG) depending on the installed capacity (§ 7 Art. 1 KWKG). The CHP remuneration includes the average price for base-load electricity (CHP-index) [38] in addition to the federal surcharges.

3. Use Case

We apply the developed model to two typical MFHs according to TABULA, which defines typical building types for residential buildings of 13 different European countries [39].

Table 4. Use case: MFH D and MFH H from TABULA.

	MFH D	MFH H
Construction period	1949–1957	1984–1994
Living area	575 ${\mathrm{m}}^2$	707 ${\mathrm{m}}^2$
Apartments	9	10
Annual heat demand	210 kWh/${\mathrm{m}}^2$	115 kWh/${\mathrm{m}}^2$
Nominal heat load	61 kW	43 kW
BOI	74.2 kW	52.0 kW
TES	45.6 kWh	56.1 kWh
Facade	1.2 W/(${\mathrm{m}}^2$K)	0.6 W/(${\mathrm{m}}^2$K)
Roof	1.6 W/(${\mathrm{m}}^2$K)	0.4 W/(${\mathrm{m}}^2$K)
Window	3.0 W/(${\mathrm{m}}^2$K)	3.0 W/(${\mathrm{m}}^2$K)

presents the specifications for both buildings, which mainly differ in the construction age and thus also in their energetic quality (e.g., heat demand and building envelope). The energy system of the initial building consists of a gas BOI with low efficiency (82%) based on data from TABULA. Since Hamburg is considered a good example in Germany for the implementation of an energy-differentiated local reference rent as described in Section 2.4.2, we choose Hamburg as the location for our study [32]. Here, local reference rent levels are determined by dividing the final energy demand into five levels (0–4). MFH D (MFH H) is classified as level 0 with a final energy demand higher than $167.7$ kWh/m² ( $84.3$ kWh/m²) and the highest level 4 is for final energy demands below $121.0$ kWh/m² ( $48.5$ kWh/m²). This results in a local reference rent dependent on the energetic level between 7.06 and 9.88 €/${\mathrm{m}}^2$ for MFH D and between 6.30 and 10.30 €/${\mathrm{m}}^2$ for MFH H, respectively.

Table 5. Energy prices from the end of 2022 (Scenario 1) and predictions for 2030 (Scenario 2) [41], and assumptions for feed-in of CHP and PV electricity.

		Scenario 1	Senario 2	Unit
Prices	Gas	0.2004	0.125	€/kWh
	Electricity	0.4007	0.327	€/kWh
	CO2	0.03	0.105	€/kg
Revenues	CHP index feed-in	0.1928		€/kWh
	CHP feed-in	0.044–0.016 *		€/kWh
	CHP self-consumption	0.015–0.08 *		€/kWh
	PV feed-in	0.082–0.109 *		€/kWh
	PV self-consumption	0.0167–0.0267 *		€/kWh

* dependent on the installed power.

provides information about energy tariffs and assumptions about revenues from feed-in electricity. General economic parameters, e.g., for energy price developments, and prices for all considered technologies are listed in Appendix A.1. For improved solving times, we apply a k-medoids clustering following Domínguez-Muñoz et al. [40] and solve the problem for six representative days with an MIP gap of 0.1%.

To derive the external influences due to ambient heat and solar irradiation, local weather data from Germany’s National Meteorological Services are applied for the location of Hamburg [42]. In this work, hourly resolved ambient temperatures and solar irradiances for average years are used. The method of Richardson et al. [43] serves as a basis for the definition of electricity demand profiles. Profiles for domestic hot water are retrieved by combining the presence profiles according to Richardson et al. with the domestic hot water profiles according to Beausoleil-Morrison [44]. In each case, the load profiles take into account that the peak demand grows degressively with the number of households, since a temporal distribution of the loads takes place.

4. Results

In the first step, we analyze how the renovation decision of the building owner differs between owner-occupied and rented buildings. Initially, we consider a scenario without any of the suggested solutions to the landlord–tenant dilemma in Section 4.1. In a second step, we investigate potential solutions to the landlord–tenant dilemma in Section 4.2. Section 4.2.1, Section 4.2.2 and Section 4.2.3 examine the impact of tenant electricity, CO₂ cost allocation, and energy-differentiated local reference rent on the landlord–tenant dilemma. In a final analysis, Section 4.2.4 evaluates the combination of all solutions. To evaluate the effects on the three problems of the landlord–tenant dilemma, for each analysis, we present the following results:

• Problem 1: Does a building owner invest in an energy system (power in kW) or the building envelope (retrofitting status)?
• Problem 2: What are the resulting total costs of the energy system, revenues of the landlord, and costs of the tenant?
• Problem 3: What are the resulting emissions of the BES?

4.1. Effects of the Landlord–Tenant Dilemma

Figure 4 compares the renovation decisions of owner-occupied buildings (OO) with the decision of a landlord of a rented building (LT) for MFH D (left) and MFH H (right). The top plot illustrates annual costs and emissions. For the owner-occupied building, it visualizes total annual costs, including capital- and operation-related costs (see Table 3). In case of the landlord–tenant relationship, the total annual costs only serve as a reference and the plot additionally includes annual incomes for the landlord and annual costs for tenants. The second plot reveals the investment decision in the energy system by presenting the power of the chosen energy supplier (BOI, ${\mathrm{CHP}}_{\mathrm{th}}$, HP, EH, STC, PV, ${\mathrm{CHP}}_{\mathrm{el}}$) and the capacity of energy storage (TES, BAT). The following two plots demonstrate the operational behavior of thermal and electrical devices. Finally, the table at the bottom shows the renovation decision on the building envelope, which ranges from 0 for no modernization to 3 for ambitious modernization.

Considering total costs and emissions, it becomes obvious that for both buildings and scenarios, the landlord’s decision for the rented building is worse than the decision in an owner-occupied building. While the landlord of the rented building does not change the energy system at all, the owner of the owner-occupied building makes changes to the energy supply system and the building envelope for both buildings and price scenarios. In all cases, the owner of the owner-occupied building chooses an HP in combination with an EH and a PV. Furthermore, the decision leads to an increased insulation of the roof (R) to level 3 in all scenarios. This leads to the highest reduction in the energy demand of the building compared to the rented building for MFH D in price scenario 1, resulting in the highest gap in terms of costs (+59.6%) and emissions (+283.0%).

The results illustrate that there is no incentive for a landlord to invest in the BES or in the building envelope under the influence of the retrofitting fee (problem 1). Moreover, our results show that for both buildings the landlord chooses an energy system that is sub-optimal in terms of total annual costs and emissions compared to the total cost optimum of an owner-occupied building (problems 2 and 3).

4.2. Solutions to the Landlord–Tenant Dilemma

Based on the depiction of the landlord–tenant dilemma, we examine solutions already used in practice (tenant electricity and CO₂ cost allocation) or proposed in the literature resp. only partly applied in a few German authorities (energy-differentiated local reference rent).

4.2.1. Solution 1: Tenant Electricity

As a first solution approach, we examine the effects of tenant electricity—a perceived incentive for landlords due to the simplified process of selling self-generated electricity to tenants. Figure 5 shows the results for the application of tenant electricity compared to the results without any solution approach (see Figure 4).

In contrast to Figure 4, the results reveal that tenant electricity leads to investment in PV in all considered scenarios. However, the owner’s decision leads to mainly fossil-based technologies for the remaining energy system. The energy system of MFH D, besides a small HP, mainly consists of a CHP as the landlord is able to sell the produced electricity to the tenants and receives money for the feed-in of CHP electricity to the grid. In the case of MFH H, the landlord decides to continue the operation of the old BOI. Moreover, the landlord does not invest in the building envelope. Consequently, the fossil-based thermal energy supply is still high and although all scenarios include a PV, total costs and emissions decrease only slightly. Compared to the results without tenant electricity, the lowest emission reduction can be observed for MFH H with 4.7% in price scenario 2 and the highest with 12.2% for MFH D in price scenario 1. A similar trend applies to the costs with reductions of 2.7% and 12.8% for the respective cases. Another positive effect is that the reduced tenant electricity price compared to a conventional price tariff also reduces tenants’ costs compared to the reference case.

Regarding the landlord–tenant dilemma, we observe an improvement in all three problems. In particular, tenant electricity incentivizes the investment of landlords in PV (problem 1). On the other hand, for the heating system, tenant electricity favors CHP rather than an HP, which is consistent with the findings of Braeuer et al. [7].

4.2.2. Solution 2: CO₂ Cost Allocation

Similar to tenant electricity, CO₂ cost allocation has already been implemented in practice in Germany. Compared to the reference case without any applied solution, the CO₂ cost creates incentives especially for investments in MFH D (see Figure 6). In MFH D, an HP combined with an EH is installed. Again, the building envelope is not renovated. Still, a more climate-friendly operation is motivated by the the CO₂ prices, resulting in a relatively high thermal coverage of the HP of around 64.8% in both price scenarios., resulting in emission reductions of 41.2% (price scenario 1) and 39.8% (price scenario 2) compared to the reference case. However, the level of the CO₂ price (price scenario 1: 0.03 €/kg, price scenario 2: 0.105 €/kg) does not influence the technology choice for MFH D. The situation is different for MFH H, where technology choices differ between the two price scenarios. While the CO₂ costs have no influence in price scenario 1, in price scenario 2 the application of CO₂ cost allocation leads to the investment in a small HP. This leads to emission reductions of 28.4% compared to the reference case. Furthermore, CO₂ cost allocation results in decreasing costs for tenants in all scenarios.

Regarding the landlord–tenant-dilemma, the results show a greater improvement for the older building (MFH D). Here, the landlord invests in an HP (problem 1) and is encouraged to keep the thermal coverage of the HP high during operation (problem 3). The costs for the tenants also decrease in the considered price scenarios (problem 2). However, for the newer building with a more energy-efficient status before renovation (MFH H), only high CO₂ prices within price scenario 2 show the same effects but to a much smaller extent.

4.2.3. Solution 3: Energy-Differentiated Local Reference Rent

Figure 7 shows the effect of an energy-differentiated local reference rent for the two buildings MFH D and MFH H and the pricing scenarios 1 and 2. We find that in all scenarios with the applied solution compared to the reference case, an electrification of the energy system takes place by choosing an HP, and in most cases an EH, instead of a BOI. This leads to reduced emissions of 37.1% for MFH D for both price scenarios and 50.7% for price scenario 1 and 45.1% for price scenario 2 for MFH H. Despite the high reduction in emissions compared to the reference case, in the case of an owner-occupied building the emissions are still 58.5% lower for MFH D in both price scenarios and 18.0% resp. 26.6% lower for MFH H in the respective scenarios (see Figure 4). In addition to the different energy systems, in the case of MFH H, the landlord also decides to renovate the roof to level 2, but still does not reach the highest energy standard unlike the owner-occupied building (see Figure 4).

In terms of costs, the results show differences in revenues for the landlord and costs for the tenants compared to the respective reference cases. Due to the energy-differentiated local reference rent, landlords can increase the rent more if a lower final energy demand is achieved. Thus, in scenarios with energy differentiation, the landlord chooses renovation measures that result in the highest level of energetic quality (level 4), which allows for the highest rent increase. This results in higher revenues for the landlord for both buildings, as well as higher costs for tenants in the case of MFH H, since the rent increase is not compensated for by lower energy costs. Within MFH H, we can observe the highest increase in revenues for the landlord in price scenario 1 (+12.4%) and the highest increase in costs for the tenants in price scenario 2 (+6.2%) compared to the reference case, where the costs for the tenants are also not in the interest of the landlord.

Regarding the landlord–tenant dilemma, we see a reduction in the severity of the first problem of lack of incentives for the landlord’s investments, especially for MFH H, e.g., investments in HP and EH and renovation of the roof. In addition, the landlord invests in climate-friendly technologies (problem 3). With regard to the second level of the dilemma, the tenant is burdened with higher costs, since the energy-differentiated local reference only takes into account the final energy demand and not the operational costs. This allows the landlord to significantly increase the rent, which is not offset by appropriate measures to reduce operating costs. Thus, we can even observe an intensification of the second problem. In the case of the older building (MFH D) though, the effects of an energy-differentiated local reference rent are only moderate.

4.2.4. Combination of All Three Solutions

Finally, Figure 8 presents the combination of the three solutions—tenant electricity, CO₂ cost allocation, and an energy-differentiated local reference rent. Again, the energy system is defossilized. Compared to the reference case, the combination of the solutions leads to the most favorable results in terms of emissions (emission reductions for MFH D of 54.1% in price scenario 1 and 58.4% in price scenario 2 and for MFH H 53.0% in both price scenarios).

In terms of cost allocation, the landlords’ revenues increase under the combined solution compared to no solution. For the costs for tenants, the results differ between the two buildings. While costs for tenants decrease slightly for MFH D (4.8% for both price scenarios), they increase for MFH H (4.0% and 4.6%). However, compared to the exclusive application of an energy-differentiated local reference rent (see Section 4.2.3), the tenant’s costs are reduced by 1.5% for MFH H in both price scenarios.

In combination, the landlord’s decision is the most favorable for a more climate-friendly solution, addressing the first and the third problems of the dilemma. However, the second level of the dilemma is only marginally improved, as the landlord’s decision still does not take operating costs into account.

5. Discussion

The results show that the existing and proposed solutions partially incentivize renovation decisions (problem 1), leading to lower emissions (problem 3) compared to the case without any of the solutions. In contrast to owner-occupied buildings, however, the incentives are not high enough to encourage renovations that contribute to significant emissions reductions (problem 1 and 3). Also, only an energy-differentiated local reference rent partly motivates investments in the building envelope (problem 1). The costs for tenants are only slightly reduced with most solutions and even increase with the application of an energy-differentiated local reference rent, as the operating costs are not directly taken into account in any of the solutions presented (problem 2).

The impact of the potential solutions for the two building types MFH D and MFH H differ. For MFH D, which represents an older typical building from the years 1949–1957, tenant electricity has the greatest impact in terms of incentives for investment in new technologies (CHP and HP), tackling problem 1. With regard to problems 2 and 3, however, the application of the CO₂ cost allocation provides the most significant potential. For MFH H, the CO₂ cost allocation also shows the best results for problem 2. However, in contrast to MFH D, the energy-differentiated local reference rent offers the greatest incentive to invest in new technologies (problem 1), which at the same time reduce emissions the most (problem 3).

5.1. Limitations and Recommendations Regarding the Solution Approaches

5.1.1. Tenant Electricity

The consideration of tenant electricity promotes investment in PV systems. In practice, however, high bureaucratic hurdles and a low level of acceptance among tenants prevent the spread of tenant electricity [45]. The tenant electricity price (which is currently limited to a maximum of 90% of the basic supply tariff) has a considerable influence on the decision of tenants whether they select the tenant electricity tariff or another tariff. The study is currently based on the assumption that all tenants will opt for tenant electricity, but this is not yet the reality and results will therefore differ in practice. In order to increase relevance and acceptance, it is therefore necessary to find an optimal price that increases acceptance while still providing sufficient incentive for landlords to invest in PV.

In addition to PV, tenant electricity also favors the installation of CHP. The consideration of CHP systems in tenant electricity concepts should be evaluated in line with the market development of CHP systems in MFHs and, if necessary, adjusted to achieve a more targeted steering effect towards photovoltaic systems and heat pumps.

5.1.2. CO₂ Cost Allocation

The CO₂ cost allocation is particularly effective for old buildings, but there are rarely incentives for actions in buildings with better insulation standards. The rising CO₂ price in scenario 2 stimulates more renovation measures, which suggests that even higher CO₂ prices will have a further steering effect. However, these are not sufficient to achieve the climate targets. In addition, the landlord is completely exempt from the costs for the lowest emission level, which is not yet compatible with a carbon-neutral building stock and therefore does not provide the necessary incentive for climate targets. In order to achieve climate neutrality by 2045, CO₂ prices and emission levels must be adjusted.

5.1.3. Energy-Differentiated Local Reference Rent

An energy-differentiated local reference rent is a promising approach, especially in combination with the two existing solutions (tenant electricity and CO₂ cost allocation), for levels 1 and 3 of the dilemma. The landlord invests in an HP mostly combined with an EH in all considered scenarios. However, measures on the building envelope are still only partially addressed, resulting in high energy demands and high flow temperatures for the heating system of the considered buildings. Due to the high level of electrification, this could lead to an overload in the power grid during peak loads in the future. With respect to the second problem of the dilemma, the energy-differentiated local reference rent leads to worse results than the reference case. In order to strengthen social compatibility and to ensure that landlords do not benefit disproportionately at the expense of tenants facing rising costs, some extensions are needed. In addition to the final energy demand, the actual energy demand should also be taken into account. This also includes the insulation standard of the building envelope. As a result, significantly higher electricity costs and also high peak loads within the electricity grid can be avoided when installing an HP. Future investment in PV systems should also be taken into account, which can be achieved by switching from energy-related to emissions-related criteria.

Another problem lies in the concept of different rent levels. Although moving up to a higher rent level generally provides incentives, the rent levels are also limited to the highest possible rent level and therefore do not encourage any further improvements. For the older building (MFH D), with an initially very high final energy demand, the highest rent level has already been reached by installing a small HP, which means that there are no further incentives for the landlord for greater investments.

5.2. Limitations of the Optimization Framework

In the optimization framework, we assume that the operation of the energy system is always optimal as we have perfect foresight, allowing, e.g., load management. This means that we likely underestimate operational costs in the MILP. This is particularly relevant for the operation of the HP in the owner-occupied building, since the operation of the HP is highly dependent on the outdoor temperature and covers the entire heat demand within the considered period (see Figure 4). This implies that in practice the results are closer to each other and the second level of the landlord–tenant dilemma might be slightly less severe than presented. On the other hand, within the landlord–tenant relation, tenants are the ones burdened with operating costs. This means an underestimation of the operational cost could also intensify the second level of the landlord–tenant-dilemma. Another limitation lies in the modeling of the flow temperatures of the heating system. As we assume a constant flow temperature of 55 °C in the heating system, we might neglect potential interactions between insulation standard and required supply temperatures. Besides the operation of the energy system, in practice, the heating behavior of the tenants might also look different. For now, all apartments are assumed to be heated to 20 °C. Future studies should consider user behavior and related effects such as rebound and prebound effects as this could have a great impact on the results [46]. In addition to user behavior, the owner is also portrayed as an economically rational decision maker. Considerations of real-world decisions should also be taken into account in the future. In addition, the results are very sensitive to the price assumptions for operation and investments in systems and the building envelope. Therefore, in addition to the two implemented price scenarios, other possible price developments should be investigated or sensitivity analyses should be carried out in order to make the results more transferable.

6. Conclusions

We extended an MILP for renovation decisions regarding the energy system and the building envelope to analyze the landlord–tenant dilemma. The dilemma consists of three levels. First, the landlord’s missing incentives to invest in renovation measures. Second, the increasing costs for tenants due to the landlord’s decisions. Third, the landlord’s lack of incentives for technologies that lead to emission reductions and are in line with climate targets.

To examine these three levels, we integrated the different perspectives and cost shares of landlords and tenants into the model and the objective function. The extension included the regulatory framework from Germany, in particular the rental law, but also, for example, subsidies. Based on this, we implemented the allocation of emission costs between landlords and tenants, tenant electricity, and an energy-differentiated local reference rent as potential solutions to the conflict. As a use case, we chose two typical buildings from the German city Hamburg with different construction ages.

The analysis without any of the proposed solutions confirmed all levels of the dilemma. The results showed that in rented buildings, the landlord has no financial incentives to invest in new technologies (problem 1). In addition, the landlord’s decisions are less favorable in terms of total costs (+6%) and emissions (+283%) than for an owner-occupied building (problems 2 and 3). The application of tenant electricity pushed investments in PV, but on the other hand, still led to a rather fossil-based heating system. The implementation of the CO₂ cost allocation proved to be beneficial for the older building, but had little impact on the newer building and only at relatively high CO₂ costs. In the case of the energy-differentiated local reference rent, we observed greater incentives for the landlord to invest in climate-friendly technologies and, in some cases, in advanced roof insulation. However, for all solutions, individually and combined, the landlord–tenant dilemma was partially improved for the first and the third problem. With regard to the second level, we found reduced cost for tenants for the application of CO₂ cost allocation and tenant electricity. In contrast, an energy-differentiated local reference rent, although promising for the other levels of the dilemma, resulted in increasing costs for the tenants. Furthermore, due to the high level of electrification and only a partial consideration of measures on the building envelope, an energy-differentiated local reference rent could lead to high peaks in the electricity grid. Thus, we deduced that the concept of energy-differentiated local reference rents needs to be extended and should not only be limited to the final energy demand.

In order to transfer these results further, additional building types and influences of different boundary conditions (e.g., location, price developments) should be investigated in the future. Furthermore, the operation of the energy system and user behavior should be modeled more precisely.